Investigating Compounds Which Preserve Freshness of Black Tulip Longer

Objective

The main goal of this study is to identify the compound(s) that should be used to preserve the freshness of black tulip by experimenting the compounds on the cut roses.

Data Description and Study Design

The data consists of total of 3540 flowers. 1770 flowers were taken from each of the gardens and types (species) with 236 flowers assigned to each of the fifteen compounds. The study was located in France, where the garden is divided into a south and north plot, both containing roses from Hybrid Tea and Floribunda species. An experimental design was constructed to select the best compound(s) that preserves the freshness of roses for a long time. The flowers were randomized using the location (North=1, South=2) and the species (Floribunda=1 and Hybrid Tea=2) as stratification factors. There were a total of four plots utilized, two north and two south with one plot containing Floribunda and the other Hybrid Tea species. In order to be included in the experiment, the cut flowers had to be of the same stage of growth. To improve the representativeness of the sample (decrease bias), to make variability similar and to reduce sampling error, the roses from each stratum were selected using simple random sampling. Sampled roses were assigned to different preservation compounds in such a way that equal amounts of flowers from each stratum were assigned to each compound in order to have a balanced design. After all roses were put in their specific treatment vase, their freshness was measured each day for 30 days. Measurements occurred according to the freshness criteria defined as: presence of color change, hanging flower stalk and angle of bending at least 10% (36 °). When none of these criteria were met, the flower was considered as fresh and a value 1 was given. When one or more criteria was met, then the flower was considered not to be fresh and a value 0 was given. After 30 days of measurement, how long each flower survived (number of days that the cut rose stayed fresh) was recorded as the response variable. In the night of day 6 to day 7, 26 roses were severely damaged/destroyed, and they could no longer be used in the experiment nor the analysis as they were considered to be missing values. Because of a shortage of rose bushes, more than one roses were needed to be cut from the same rose bush. This is indicated by the variable Bush.

Sample Size Determination

The sample size of the study is obtained using R version 3.1.1 and is based on analysis of variance. The power was taken to be 85% with the level of significance 5%. The effect size was stated as 1 day. A pilot study was done to estimate the variance parameters. This pilot study was conducted for 60 roses stratified for each species and location for each compound. Based on all the obtained and chosen parameters, the calculated total sample size was 3540 flowers with 236 flowers per compound.

Explanatory Data Analysis

Explanatory data analysis was used to get an insight into the data set, explore various structures, relationships and patterns. Summary statistics and several types of plots were used to describe the data.

Statistical Models

Analysis of Variance (ANOVA) Three-way ANOVA model was used to determine if there is an effect of the three factors (compound, garden and type) on a continuous dependent variable (number of days of freshness). If the overall test of factor(s) are found to be significant, how the levels of factor differ should be examined in more detail in order to select the best compound(s). The mathematical description of the model

$$Y_{ijki} = \mu \ldots + \alpha_i + \beta_j + Y_k + e_{ijkl} \quad (1)$$

where; $i = 1, \ldots, 15; j = 1, 2; k = 1, 2; l = 1, \ldots, ni; Y_{ijki}$ is the number of days of freshness of $l^{th}$ flower from $j^{th}$ specie and $k^{th}$ garden using compound $i$, $\alpha_i$ is the $i^{th}$ compound effect with restriction $\sum_{i=1}^{15} \alpha_i = 0$, $\beta_j$ is the $j^{th}$ specie effect with restriction $\sum_{j=1}^{2} \beta_j = 0$ and is the $k^{th}$ garden effect with restriction $\sum_{k=1}^{2} Y_k = 0$, eijkl is the error term.

**Generalized Estimating Equations (GEE)**

Since the response variable, the number of days of freshness, is a count data and roses are clustered within bush, GEE based on Poisson regression model is considered as an alternative to the ANOVA model. The Generalized Estimating Equations approach, which was introduced by Liang KY [8] is a method for analyzing correlated outcome of the data. The within-subject correlation among the repeated measures is taken into account by defining a working correlation structure and fitting that structure into the estimation [9]. The method provides two estimates of the standard errors: the model-based (from a naively specified working assumption) and the empirical ones (robust). When adopting GEE, one does not use information of the correlation structure to estimate the main effect parameters. As a result, it can be shown that GEE yields consistent main effect estimators, even when the correlation structure is miss-specified [10]. The choice of the working assumption about the correlation structure can be made from independence, exchangeability, unstructured and auto-regressive AR (1).

In this study, exchangeable assumption was chosen since the correlation between bushes are not equally spaced measurements and do not depend on time. When the model-based and empirical standard errors are far apart, this can be seen as an indication for a poor choice of working assumptions. A poor working assumption is not wrong, but may hamper efficiency and, when at all possible, it may be of interest to then try alternative working assumptions [10].

The mathematical description of the model is given as:

$$\log(\mu_i) = \alpha_1 + \alpha_2 c_2 + \ldots + \alpha_{15} c_{15} + \alpha_{16} \text{Type} + \alpha_{17} \text{Garden} (2)$$

Where, $Y_i = \text{Poisson}(\mu_i)$, $C_2$, $C_3$, ..., $C_{15}$ are dummy variables for compound 2, ..., 15, respectively, $Y_i$ is the outcome for rose i and $\mu_i$ is the expected number of days $j^{th}$ rose stays fresh.

**Generalized Linear Mixed Models (GLMM)**

In this study, more than one roses were taken from a single bush and freshness of a flower was measured on a daily basis. As such, the measurements were taken from a single flower at different times and roses from the same bush were assumed to be correlated. Hence, it is advisable to use a longitudinal data analysis method to handle the correlated nature of the data. The response variable, freshness of flowers, is binary data but the measurements are correlated, therefore the common binary logistic regression model is not appropriate [11]. Therefore, GLMM was applied to select the compound with the highest probability of keeping the cut rose flowers fresh.

The mathematical description of the model is as follows:

$$\logit(\pi_{ijk}) = \alpha_6 + \text{Bush}_k + \alpha_1 \text{Day}_j + \alpha_2 c_2 \text{Day}_j + \ldots + \alpha_{15} c_{15} \text{Day}_j + \alpha_{16} \text{Type} + \alpha_{17} \text{Garden} (3)$$

Where; is binary outcome for rose i from bush k at day j, is the probability of the $j^{th}$ rose from $k^{th}$ bush still being fresh at $j^{th}$ day, $C_{i2}, \ldots, C_{i15} = 1$ if compound 2, ..., 15 is used, else 0 respectively, Type=1 if Type1 is used, else 0 and Garden=1 if Garden1 is used, else 0. Distilled water for compound, Hybrid tea for type and South garden for garden are considered as reference categories in all models considered in this study.

**Software**

Statistical analysis are performed using SAS version 9.4 and R version 3.1.1. All tests were performed at significance level of 5%.

**Exploratory Data Analysis**

The data consisted of 3540 rose flowers with 26 missing and 3514 effective flowers. 1770 flowers were taken from each of the gardens and types (species). From the summary statistics provided in Appendix 1, Table 7 (part 7.1), it was observed that compound6 and compound12 have the highest average number of days of freshness; 21.12 and 20.97 with a standard deviations of 4.43 and 4.33, respectively while distilled water had 8.25 days with a standard deviation of 2.96. A box plot of the number of days of freshness against each compound was given in Figure 1. Compound6 and compound12 seems to have the highest average number of days of freshness while compound2 and compound10 appears to be providing the lowest number of days of freshness.

Figure 2 below represents the number of days of freshness against each of the compounds within the gardens. Compound6 and compound12 have the highest number of days of freshness while compound2 and compound10 have the lowest number of days of freshness in both of the gardens (1=North and 2=South). Additionally, it can be observed that there is no interaction between compound and garden. It can also be seen that garden2 has slightly higher preserving power than garden1.

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Figure 3 presents the number of days of freshness against each compound within the types (species). It has also observed that compound6 and compound12 have the highest number of days of freshness while compound2 and compound10 have the lowest number of days of freshness in both of the types (1=Floribunda and 2=Hybrid Tea). Besides, it is observed that there is no interaction between compound and type, and that type2 has slightly higher surviving power than type1. That being said, a rose from the species of Hybrid Tea may preserve its freshness for a longer time than a rose from Floribunda.

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ANOVA Model

Three way ANOVA model was used with factors compound, garden and type with fifteen, two and two levels, respectively. The main factors compound, garden and type have significant effect in the freshness of the roses since the p-value for all of them in the F-statistics presented in Table 1(part 1.1) are very small (<0.0001) in comparison to 5% significance level. However, as their interactions were found to be insignificant, the model was refitted without

From the results in Table 2, it was observed that all the fourteen compounds have significantly different effect in comparison to compound1 (reference) since the p-value for all fourteen compounds are smaller than 5% significance level. All compounds, which have negative average estimated values have less preserving ability than the distilled water while those with positive average estimated values have higher preserving ability for the cut rose than distilled water.

It can be observed that the compound6 has a mean estimate of approximately 12.88 higher than the distilled water in preserving the freshness of cut roses. Compound12 is the second best with a mean estimate of approximately 12.74 higher than the distilled water. This shows that compound6 and compound12 both have almost the same and highest preserving ability. Compound10 and compound2 have lower preserving power for the cut flowers than distilled water.

Since the p-value is small, there is significant difference between the gardens (1=Northern and 2=Southern). Garden2 has the highest power in preserving the cut rose, which means that if a rose is from southern part of the garden, it can survive for a longer time than a rose taken from northern part. Since the p-value for the type (specie) is small, there is significant difference between the two species (1=Floribunda and 2=Hybrid Tea). As it is observed from Table 2, Type2 has higher surviving ability such that arose from Hybrid Tea specie has higher surviving ability than a rose from Floribunda specie. In the ANOVA model, there are assumptions that should be fulfilled. Accordingly, the residuals of the model are assumed to follow a normal distribution with mean zero and constant variance. However, from the results of Kolmogorov-Smirnov normality test and Brown and Forsythe’s homogeneity of variance test, both the normality of error terms as well as the constancy of variance are found to be violated (Appendix, Table 6 and Figure 4).

Since the normality and constancy of variance assumptions are violated, Box-Cox transformation was done using a value of 0.5 λ = (Appendix, Figure 5). However, still both assumptions were violated (Appendix, Table 6). Outliers were identified by the Bonferroni test and observations with absolute studentized deleted residuals > ( ) 1 0.40

2 3514 , 3496

$$ \left[ t \left(\frac {1 - \alpha}{2 (3 5 1 4) , 3 4 9 6}\right) \right] = 0 $$ − are considered to be outliers while outliers with DFFIT > 2 17 2 2&( ) 0.212 1514  =     are considered to be influential. Although, there are outliers observed, there are no influential outliers.

It should be noted that since the assumptions of the ANOVA model are not fulfilled, the conclusion based on ANOVA may not be sufficient enough.

Generalized Estimating Equations (GEE)

The assumptions in the ANOVA model were not met even after transformation of the response variable was done. Therefore, Generalized Estimating Equation (GEE) based on Poisson model was applied as an alternative to ANOVA. Due to the fact that more than one roses are taken from a bush, the roses within the same bush are likely to be correlated. For this reason, GEE was fitted as a model to account this within correlation structure.

In this study, GEE is used by assuming exchangeable correlation between roses within the same bush, i.e.

( ) ,

corr Yij Yik

$$ = \alpha \text {f o r} j \neq k \text {a n d} \operatorname {c o r r} \left(Y i j, Y i k\right) = 1 $$

corr Yij Yik =

$$ \begin{array}{l} \operatorname {c o r r} (1 i j) \\ \mathrm {f o r} j = k. \\ \end{array} $$

. The score statistics type III results provided in

Table 1(part 1.2) shows that since the p-value for compound,

type, and garden are less than 5% level of significance, those

variables have a significant effect on the mean number of

days cut flowers stay fresh.

A parameter estimate based on empirical standard error are given in Table 3. All compounds are highly significant, indicating that all compounds preserve the cut roses significantly different from distilled water since the p-value for all compounds are less than 5% level of significance. The parameter estimates of compound2 and compound10 are negative, leading to the conclusion that compound2 and 10 preserve a cut rose shorter time than distilled water. The other 12 compounds kept the flower fresh longer than distilled water with the positive parameter estimates. The results of the estimated mean number of days a cut rose stayed fresh is given in Appendix, Table 7(part 7.2). The highest mean number of days of freshness for a cut rose was obtained using compound6 and 12 with 21.1 days and 21.0 days, respectively. Similarly the lowest mean number of days of freshness of a cut rose was obtained using Compound2 and Compound10 with 5.6 days and 5.5 days, respectively. The mean number of days of freshness of roses using distilled water was obtained to be 8.3 days. It is also noted that roses from garden2 and of type2 stayed fresh for a longer period of time in comparison to garden1 and type1, respectively.

In addition, the empirical standard error estimates are almost approximately the same as the model based on standard error estimates shown in Table 4. This implies that the assumed covariance or correlation structure is correct, as suggested by Muhlenberg’s and Verbeke [10].

Generalized Linear Mixed Model (GLMM)

In this study, more than one rose was taken from a single bush and freshness of a flower was measured on a daily basis. As such, the measurements taken from a single flower at different times and roses from the same bush are correlated. Due to cluster of roses within a bush and the measurements being taken repeatedly over time, it is advisable to use a longitudinal data analysis method to handle the correlated nature of the data. Since our response variable, the freshness of a flower, is binary data and the measurements are correlated, the common binary logistic regression model is not appropriate. Therefore, GLMM was applied to select the compound with the highest probability of keeping the cut rose flowers fresh. The results of generalized linear mixed model provided in Table 5 (part 5.2) shows that days, days*compound, type and garden have significant effect on the log odds of freshness

These findings are further confirmed by the predicted probability plot shown in Appendix, Figure 6. It is observed that compound6 and 12 are the best compounds in preserving the freshness of cut rose the longest since their curves are above the curves of other compounds.

As it can be seen from the results of GLMM in Table 5 (part 5.1), cut rose from garden2 (southern part) stays fresh for longer time than a rose from garden1 (northern part).

The rose of type2 (Hybrid tea) has also higher probability of staying fresh for longer time than a rose of type1 (Floribunda).